Approximation method for series expansion of input function and system using the same

ABSTRACT

An approximation method and system for series expansion of functions include the steps and functions, respectively, of: expanding an input function in Taylor series up to an (N−1)-th term; expanding the input function in Taylor series up to the N-th term; multiplying the expanded result up to the (N−1)-th term by a predetermined weight α; combining the expanded result up to the (N−1)-th term, multiplied by α, and the expanded result up to the N-th term; and dividing the combined result by (α+1).

CLAIM OF PRIORITY

[0001] This application makes reference to, incorporates the sameherein, and claims all benefits accruing under 35 U.S.C. §119 from myapplication APPROXIMATION METHOD FOR SERIES EXPANSION OF INPUT FUNCTIONAND SYSTEM USING THE SAME filed with the Korean Industrial PropertyOffice on 30 Apr. 2003 and there duly assigned Serial No. 2003-27747.

BACKGROUND OF THE INVENTION

[0002] 1. Technical Field

[0003] The present invention relates to an approximation method forseries expansion of an input function and a system using the same, and,more particularly, to an approximation method and system for seriesexpansion of an input function, which minimizes approximation errors inthe series expansion of the input function of an arithmetic type.

[0004] 2. Related Art

[0005] Generally, system design starts from upper level algorithmdesign, in which analysis and verification of the algorithm isperformed, and is followed by lower level hardware design. In order toclosely implement arithmetic functions used in the upper level algorithmdesign in the lower level hardware design, corresponding function valuesshould be stored in a memory. In this case, since function values forevery input variable should be reserved and stored, the memory shouldhave a large capacity, resulting in a heavy burden in hardware design.

[0006] Accordingly, in system design, arithmetic functions for inputvariables are typically obtained by series expansion, rather thanstoring every function value for each input variable.

[0007] However, in practice, a series expansion formula cannot beimplemented by a combination of an infinite number of terms to reach anominal function value. Instead, a series expansion approximationformula, expanded using a finite number of terms, is typically used toobtain an approximate function value. In this case, an approximationerror, or a truncation error, which represents a difference between thenominal function value and the approximated function value, occursinevitably. Such an approximation error often causes criticalmalfunction or degradation of the system. A series expansion with asmany terms as possible can reduce approximation errors, but it willautomatically lead to a system of increased complexity required toimplement such a series expansion.

[0008] Let us now assume that an arithmetic function S(x) is expressedin an infinite series expansion with terms, ƒ₀(x),ƒ₁(x),ƒ₂(x) asfollows: $\begin{matrix}{{S(x)} = {{\sum\limits_{n = 0}^{\infty}{f_{n}(x)}} = {{f_{0}(x)} + {f_{1}(x)} + {f_{2}(x)} + \cdots}}} & (1)\end{matrix}$

[0009] For example, familiar trigonometric functions often employed indigital system design can be expanded in Taylor series about x=0. Inthis case, the terms of the trigonometric functions have alternatingsigns (+, −). Some common trigonometric functions expressed using Taylorseries are as follows: $\begin{matrix}{{\cos (x)} = {{\sum\limits_{n = 0}^{\infty}{\left( {- 1} \right)^{n}\frac{x^{2n}}{2{n!}}}} = {1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots}}} & (2) \\{{\sin (x)} = {{\sum\limits_{n = 0}^{\infty}{\left( {- 1} \right)^{n}\frac{x^{{2n} + 1}}{\left( {{2n} + 1} \right)!}}} = {x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots}}} & (3) \\{{{\arctan (x)} = {{\sum\limits_{n = 0}^{\infty}{\left( {- 1} \right)^{n}\frac{x^{{2n} + 1}}{{2n} + 1}}} = {x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \cdots}}}\quad} & (4) \\{\quad {{- 1} \leq x \leq 1}} & \quad\end{matrix}$

[0010] However, in a fixed-point digital system, series expansion has afinite number of terms, thereby causing an approximation error. Forexample, if a partial sum,${S_{N}(x)} = {\sum\limits_{n = 0}^{N}{\left( {- 1} \right)^{n}{f_{n}(x)}}}$

[0011] is used to approximate the function,${{S(x)} = {\sum\limits_{n = 0}^{\infty}{\left( {- 1} \right)^{n}{f_{n}(x)}}}},$

[0012] the approximation error is expressed as E_(N)(x)=S(x)−S_(N)(x).Since the absolute value |E_(N)(X)| of the approximation error E_(N)(X)increases with the absolute value |x| of the input variable x, moreterms in the series expansion are required to reduce the approximationerror.

[0013] To reduce the approximation error in an alternating seriesexpansion, the Euler approximation method is frequently used. In theEuler approximation method, the same series expansion formula as in theTaylor series is applied up to term (N−1), while an Euler transformationformula is applied for the rest of the terms. For example, the functionS(x) presented in equation (1) can be expressed as follows in equation(5): $\begin{matrix}\begin{matrix}{{{\sum\limits_{n = 0}^{\infty}{\left( {- 1} \right)^{n}f_{n}}} = {f_{0} - f_{1} + f_{2} - \cdots - f_{N - 1} + {\sum\limits_{n = 0}^{\infty}{\frac{\left( {- 1} \right)^{n}}{2^{n + 1}}\left\lbrack {\Delta^{n}f_{N}} \right\rbrack}}}},} \\{{{{for}\quad N} = {even}}}\end{matrix} & (5)\end{matrix}$

[0014] wherein, Δ is called a forward difference operator and has thefollowing characteristics:

Δƒ_(N)≡ƒ_(N+1)−ƒ_(N)

Δ²ƒ_(N)≡ƒ_(N+2)−2ƒ_(N+1)+ƒ_(N)   (6)

Δ³ƒ_(N)≡ƒ_(N+3)−3ƒ_(N+2)+3ƒ_(N+1)−ƒ_(N), etc.

[0015] The Euler approximation method, however, also has a problem inthat the rightmost term should have a large number of terms to achievean accurate approximation and it works only for N=even.

[0016] Therefore, there is a need to provide an efficient approximationmethod for series expansion of functions and a system therefor, whichcan minimize an approximation error and be simply implemented inpractice.

SUMMARY OF THE INVENTION

[0017] The present invention provides an approximation method for seriesexpansion of functions, and a system therefor, in which an approximationerror of a nominal function value is efficiently minimized.

[0018] According to an aspect of the present invention, there isprovided an approximation method for a series expansion of an inputfunction with a finite number of terms N to minimize approximationerror, comprising the steps of: expanding the input function in Taylorseries up to an (N−1)-th term; expanding the input function in Taylorseries up to an N-th term; multiplying a predetermined weight α and theexpanded result up to the (N−1)-th term; and combining the expandedresult up to the (N−1)-th term, multiplied by α, and the expanded resultup to the N-th term; and dividing the combined result by (α+1).

[0019] It is preferable that a be greater than 0 and less than or equalto 1.

[0020] According to another aspect of the present invention, there isprovided an approximation method for a series expansion of an inputfunction with a finite number of terms N to minimize approximationerror, comprising the steps of: expanding the input function in Taylorseries up to an (N−1)-th term; multiplying an N-th term of the seriesexpansion of the input function and a predetermined weight value; andcombining the expanded result up to the (N−1)-th term and the multipliedN-th term to be an approximation function ƒ for the series expansion ofthe input function.

[0021] It is preferable that the predetermined weight value be$\frac{\left( {- 1} \right)^{N}}{\left( {\alpha + 1} \right)}$

[0022] for 0<α≦1.

[0023] It is preferable that α, obtained for corresponding respective N,be selected to minimize a maximum of the absolute approximation error.

[0024] It is preferable that α be obtained by: (a) selecting a minimuminput in a given input x area; (b) calculating ƒ for the input with thefinite number of terms N; (c) obtaining and storing an error|E_(N),_(x)| by subtracting ƒ from a nominal function value of the inputx; (d) determining whether the input x has reached a maximum value inthe given input x area, adding a predetermined increment ξ to x if x hasnot yet reached the maximum value, and repeating steps from (b), (c) and(d); (e) if x has reached the maximum value, selecting a maximum errorvalue among all the stored errors of E_(N),_(x) for all inputs; and (f)searching a to minimize the maximum error value, and storing α as theweight value for corresponding N.

[0025] According to yet another aspect of the present invention, thereis provided an approximation method for a series expansion of an inputfunction with a finite number of terms N to minimize an approximationerror, comprising the steps of: dividing a whole input area into severalpredetermined sub-intervals: expanding the input function in Taylorseries up to an (N−1)-th term in each of the sub-intervals; multiplyingan N-th term of the series expansion of the input function with apredetermined first weight for inputs on left side from a center of eachof the sub-intervals, and with a predetermined second weight for inputson right side from the center of each of the sub-intervals; andcombining the expanded result up to the (N−1 )-th term, and themultiplied N-th term with the predetermined first and second weights tobe an approximation of the input function in each of the sub-intervals.

[0026] It is preferable that the predetermined first and second weightson the left and right side in each of the sub-intervals be selected tominimize a maximum error between the approximation of the input functionwith the finite number of terms N and a nominal value of the inputfunction over all the inputs in a corresponding sub-interval.

[0027] According to still another aspect of the present invention, thereis provided a method to compensate a carrier frequency offset in anorthogonal frequency division multiplexing (OFDM) system, comprising thesteps of: estimating the carrier frequency offset {circumflex over (ε)}by using a series expansion of an arctangent function arctan(x); usingthe estimated offset to obtain a phase rotation value for a first inputsample of k=1, wherein sin(2π{circumflex over (ε)}) andcos(2π{circumflex over (ε)}) are series-expanded to minimize anapproximation error; using a phase rotation value for a previous inputsample including k=1 to obtain a phase rotation value for subsequentinput sample; and compensating the phase rotation values for all theinput samples.

[0028] It is preferable that the estimated carrier frequency offset{circumflex over (ε)} be represented by,$\hat{ɛ} = {\frac{1}{2\quad \pi}\arctan \left\{ \frac{\sum\limits_{i = 1}^{L}{{Im}\left( {{y\left( {- i} \right)}y*\left( {L - i} \right)} \right)}}{\sum\limits_{i = 1}^{L}{{Re}\left( {{y\left( {- i} \right)}y*\left( {L - i} \right)} \right)}} \right\}}$

[0029] where Re and Im represent a real part and an imaginary part of acomplex number, respectively, y(i) represents an i-th received sample, Lis a fast fourier transformation (FFT) size, and {circumflex over (ε)}is an estimated and normalized carrier frequency offset of Δ{circumflexover (ƒ)}T, where T is OFDM symbol duration.

[0030] It is preferable that the phase rotation value for the k-thsample be calculated by $\begin{matrix}{{{{For}\quad k} = 1},{{\cos \left( {\Delta \quad \hat{\omega}T_{s}} \right)} = {\sum\limits_{n = 0}^{N}{\left( {- 1} \right)^{n}\frac{\Delta \quad \hat{\omega}T_{s}^{2n}}{\left( {2n} \right)!}}}}} \\{\quad {{\sin \left( {\Delta \quad \hat{\omega}T_{s}} \right)} = {\sum\limits_{n = 0}^{N}{\left( {- 1} \right)^{n}\frac{\Delta \quad \hat{\omega}T_{s}^{({{2n} + 1})}}{\left( {{2n} + 1} \right)!}}}}} \\{{{{For}\quad k} \geq 2},{{\cos \left( {k\quad \Delta \quad \hat{\omega}T_{s}} \right)} = {\cos \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} + {\Delta \quad \hat{\omega}T_{s}}} \right)}}} \\{\quad {= {{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}T_{s}} \right)}} -}}} \\{\quad {{\sin \left( {\left( {k - 1} \right)\quad \Delta \quad \hat{\omega}T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}} \\{\quad {{\sin \left( {k\quad \Delta \quad \hat{\omega}T_{s}} \right)} = {\sin \left( {{\left( {k - 1} \right)\Delta \hat{\omega}T_{s}} + {\Delta \quad \hat{\omega}T_{s}}} \right)}}} \\{\quad {= {{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}T_{s}} \right)}} +}}} \\{\quad {{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}T_{s}} \right)}}}\end{matrix}$

[0031] According to still another aspect of the invention, there isprovided an approximation system for carrying out a series expansion ofan input function with a finite number of terms N to minimize anapproximation error, comprising an operational processing unit whichexpands the input function in Taylor series up to an (N−1)-th term,expands the input function in Taylor series up to an N-th term,multiplies a predetermined weight α and the expanded result up to the(N−1)-th term, combines the expanded result up to the (N−1)-th term,multiplied by α, and the expanded result up to the N-th term, anddivides the combined result by (α+1).

[0032] It is preferable that α be greater than 0 and less than or equalto 1.

[0033] It is preferable that α obtained for a corresponding respective Nbe selected to minimize a maximum approximation error.

[0034] According to a further aspect of the invention, there is providean approximation system for carrying out a series expansion of an inputfunction with a finite number of terms N to minimize an approximationerror, comprising an operational processing unit which expands the inputfunction in Taylor series up to an (N−1)-th term, multiplies an N-thterm of the series expansion function and a predetermined weight value,combines the expanded result up to the (N−1)-th term and the multipliedN-th term to be an approximation function ƒ for the series expansionfunction.

[0035] It is preferable that the predetermined weight value be$\frac{\left( {- 1} \right)^{N}}{\left( {\alpha + 1} \right)}$

[0036] for 0<α≦1.

[0037] It is preferable that a obtained for a corresponding respective Nbe selected to minimize a maximum approximation error.

[0038] It is preferable that a be obtained by (a) selecting a minimuminput in a given input x area; (b) calculating ƒ for the input withfinite number of terms N; (c) obtaining and storing an error E_(N,x) bysubtracting ƒ from an nominal function value of the input x; (d)determining whether the input x has reached a maximum value in the giveninput x area, adding a predetermined increment ξ to x if x has not yetreached the maximum value, and repeating steps (b), (c) and (d); (e) ifx has reached the maximum value, selecting a maximum error value amongall the stored errors of E_(N,x) for all inputs; and (f) searching α tominimize the maximum error value, and storing α as the weight value fora corresponding N.

[0039] According to another aspect of the invention, there is providedan orthogonal frequency division multiplexing (OFDM) system forcompensating a carrier frequency offset, comprising: an estimator forestimating the carrier frequency offset {circumflex over (ε)} by using aseries expansion of a function arctran(x); a first phase rotationcalculator for using the estimated offset to obtain a phase rotationvalue for a first input sample of k=1, wherein sin(2π{circumflex over(ε)}) and cos(2π{circumflex over (ε)}) are series-expanded to minimizean approximation error; a second phase rotation calculator for using aphase rotation value for a previous input sample including k=1 to obtaina phase rotation value for subsequent input sample; and a compensatorfor compensating the phase rotation values for all the input samples.

[0040] It is preferable that the estimated carrier frequency offset{circumflex over (ε)} be represented by${\hat{ɛ} = {\frac{1}{2\quad \pi}\arctan \left\{ \frac{\sum\limits_{i = 1}^{L}{{Im}\left( {{y\left( {- i} \right)}y*\left( {L - i} \right)} \right)}}{\sum\limits_{i = 1}^{L}{{Re}\left( {{y\left( {- i} \right)}y*\left( {L - i} \right)} \right)}} \right\}}},$

[0041] where Re and Im represent a real part and an imaginary part,respectively, of a complex number, y(i) represents an i-th receivedsample, L is a fast fourier transformation (FFT) size, and {circumflexover (ε)} is an estimated and normalized carrier frequency offset ofΔƒT.

[0042] It is preferable that the phase rotation value for the k-thsample be calculated by $\begin{matrix}{{{{For}\quad k} = 1},{{\cos \left( {\Delta \quad \hat{\omega}T_{s}} \right)} = {\sum\limits_{n = 0}^{N}{\left( {- 1} \right)^{n}\frac{\Delta \quad \hat{\omega}T_{s}^{2n}}{\left( {2n} \right)!}}}}} \\{\quad {{\sin \left( {\Delta \quad \hat{\omega}T_{s}} \right)} = {\sum\limits_{n = 0}^{N}{\left( {- 1} \right)^{n}\frac{\Delta \quad \hat{\omega}T_{s}^{({{2n} + 1})}}{\left( {{2n} + 1} \right)!}}}}} \\{{{{For}\quad k} \geq 2},{{\cos \left( {k\quad \Delta \quad \hat{\omega}T_{s}} \right)} = {\cos \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} + {\Delta \quad \hat{\omega}T_{s}}} \right)}}} \\{\quad {= {{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}T_{s}} \right)}} -}}} \\{\quad {{\sin \left( {\left( {k - 1} \right)\quad \Delta \quad \hat{\omega}T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}} \\{\quad {{\sin \left( {k\quad \Delta \quad \hat{\omega}T_{s}} \right)} = {\sin \left( {{\left( {k - 1} \right)\Delta \hat{\omega}T_{s}} + {\Delta \quad \hat{\omega}T_{s}}} \right)}}} \\{\quad {= {{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}T_{s}} \right)}} +}}} \\{\quad {{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}T_{s}} \right)}}}\end{matrix}$

BRIEF DESCRIPTION OF THE DRAWINGS

[0043] A more complete appreciation of the invention, and many of theattendant advantages thereof, will be readily apparent as the samebecomes better understood by reference to the following detaileddescription when considered in conjunction with the accompanyingdrawings in which like reference symbols indicate the same or similarcomponents, wherein:

[0044]FIG. 1 is a graph showing approximation error characteristics ofan arctangent function arctan(x), expanded in Taylor series with afinite number N of terms;

[0045]FIG. 2 is a schematic block diagram for implementing anapproximation method for series expansion according to the presentinvention;

[0046]FIG. 3A˜3C are graphs showing approximation error characteristicsof sin(x), cos(x), arctan(x), respectively, according to various valuesof weight α, according to the present invention;

[0047]FIG. 4 is a flowchart for calculating the weight α according tothe present invention.

[0048]FIG. 5A is a graph comparing approximation error characteristic ofarctan(x) at N=7, expanded according to Taylor series expansion, Eulerapproximation method, and an approximation method of the presentinvention;

[0049]FIG. 5B is a graph comparing approximation error characteristic ofarctan(x) at N=8, expanded according to Taylor series expansion, Eulerapproximation method, and an approximation method of the presentinvention;

[0050]FIG. 6A shows an approximation error characteristic of functionarctan(x), the input area [−1, 1] of which is divided into 4 intervals,each of which is centered at x=−0.75, x=−0.25, x=0.25 and x=0.75,respectively, and separately expanded according to the present inventionin each of the 4 intervals;

[0051]FIG. 6B shows two comparable absolute approximation errorcharacteristics of function arctan(x) according to Taylor seriesexpansion and an approximation method of the present invention, whereinfour terms are used, and wherein inputs in both cases are divided intoseveral intervals in which the arctangent function is separatelyexpanded;

[0052]FIG. 7A˜7C are constellation diagrams of the Quadrature AmplitueModulation-Orthogonal Frequency Division Multiplexing (QAM-OFDM) for 10OFDM symbols according to conventional series expansions and the presentinvention, respectively.

DETAILED DESCRIPTION OF THE INVENTION

[0053] The present invention provides an approximation method for seriesexpansion of functions and a system therefor.

[0054]FIG. 1 is a graph showing approximation error characteristics,E_(N)(X), of an arctangent function arctan(x) implemented by Taylorseries expansion with various finite number of terms N. From FIG. 1, itcan be noted that the approximation performance is better for inputsnear x=0. In FIG. 1, for example, when N=9 and x=−1, it is shown thatthe approximation error is about 1.43 degree. This error may cause acritical system malfunction depending on the system characteristics.

[0055] From the approximation error characteristics in FIG. 1, the signof E_(N)(X) for an even and odd number of terms alternates at a given x.This is a common phenomenon for the approximation error characteristicsof all the alternating series functions, such as cos(x), sin(x)functions, presented in equations (2) and (3). Table 1 shows theE_(N)(X) sign characteristics of trigonometric functions sin(x), cos(x),and arctan(x). TABLE 1 Function name sin(x) cos(x) arctan(x) N statusodd Even odd even odd Even x > 0 − + − + − + x < 0 + − − + + −

[0056] The trigonometric functions in Table 1 are convergent whenapproximated in series expansion, so that$0 < \frac{{E_{N}(x)}}{{E_{N - 1}(x)}} \leq 1$

[0057] and sgn[E_(N)(x)]=sgn{E_(N−1)(x)}). In other words, the signs ofE_(N)(X) for even and odd numbers of terms are opposite to each other.By using these characteristics, a new approximation formula for seriesexpansion may be provided as follows: $\begin{matrix}{{{\overset{\sim}{S}}_{N}(x)} = {\frac{1}{\left( {\alpha + 1} \right)}\left( {{\alpha \quad {S_{N - 1}(x)}} + {S_{N}(x)}} \right)}} & (7)\end{matrix}$

[0058] For a given function, the approximation error of the functioncorresponding to a partial sum of (N−1) expansion terms, i.e.,S_(N−1)(X), is greater than one having the approximation error of thesame function corresponding to sum of N expansion terms, i.e., S_(N)(X),and the signs of the approximation errors are opposite to each other. Toreduce the approximation error of S_(N−1)(x), a weight α(0<α≦1) can begiven to S_(N−1)(X), while a weight of I is given to S_(N)(X). Inequation (7), the new approximation formula represents a combination ofthe partial sum of (N−1) expansion terms multiplied by the weight α andthe sum of N expansion terms, averaged by α+1. Further, based on thegeneral formula of alternating series,S_(N)(x)=S_(N−1)(x)+(−1)^(N)ƒ_(N)(x), a new approximation formula can beobtained from equation (7) as follows: $\begin{matrix}{{{\overset{\sim}{S}}_{N}(x)} = {{S_{N - 1}(x)} + {\frac{\left( {- 1} \right)^{N}}{\left( {\alpha + 1} \right)}{f_{N}(x)}}}} & (8)\end{matrix}$

[0059] Consequently, in the new approximation formula, where the finitenumber of expansion terms is assumed to be equal to N, the generallyknown series expansion formula is used for (N−1) terms, while$\frac{\left( {- 1} \right)^{N}}{\left( {\alpha + 1} \right)}$

[0060] is multiplied by an N-th term, thereby producing an efficientimproved series expansion formula that reduces the approximation error.Equation (8) is simpler than equation (5) which uses an Eulerapproximation method, and is applicable not only to functions having analternating series characteristic, but also to functions with generalseries expansion having convergence characteristic.

[0061]FIG. 2 is a schematic block diagram for implementing theapproximation method for series expansion as described above. Thefunctions of the block diagram can be implemented in an operationalprocessing unit (not shown), such as a Digital Signal Processor (DSP).

[0062]FIG. 3A˜3C are graphs showing approximation errors E_(N)(x) ofsin(x), cos(x), arctan(x), respectively, according to various values ofweight α, when the approximation is performed using equation (8).Herein, the number of terms N is set, for example to 3, and the optimalvalue of a is obtained by searching for a value which minimizes themaximum of |E_(N)(x)|, the search being performed for 0<α≦1, asrepresented in equation (9) as follows:

min{max[|E_(N)(x)|}  (9)

[0063] In general, the optimal α depends on the number of terms N andthe type of function to be expanded in series.

[0064]FIG. 4 is a flowchart for obtaining a weight α for a given N tominimize the maximum approximation error in a predetermined interval ofinput x.

[0065] A minimum value x_(min) in the predetermined interval is selectedin step 402.

[0066] By using equation (8), an approximation function ƒ=ƒ_(N,x)(N,x)of a given function is calculated in step 404.

[0067] An approximation error value E_(N,x) is obtained and stored bysubtracting ƒ from a nominal function value F corresponding seriesexpansion with an infinite number of terms in step 406.

[0068] If it is determined that present x is the maximum value withinthe predetermined interval in step 408, the method proceeds to step 412.If not, an accurately chosen increment ξ is added to x in step 410 andthe method goes back to step 404. Steps 404 thru 410 are repeated untilx is equal to or greater than the maximum value x_(max) in thepredetermined interval.

[0069] If, in step 408, it is determined that x is equal to or greaterthan the maximum value x_(max), in step 412 a maximum absoluteapproximation error is selected from among all of the approximationerror values of |E_(N,x)| for all x's, stored in step 406. The selectedmaximum absolute approximation error may be a function of weight α.Therefore, in step 414, a value of α to minimize the maximum absoluteapproximation error value can be calculated.

[0070] In step 416, the value of α calculated in step 414 is stored in amemory of the system which implements this approximation method.

[0071] When this approximation algorithm, based on equation 8, isimplemented, α is set in advance depending on a pre-determined N setthrough computer simulation. An approximation error of a functionexpanded in Taylor series using equation (8), based on the block diagramof the present invention as shown in FIG. 2, by using the optimal weightvalue α in equation (8), is effectively reduced, compared with theapproximation error obtained using the conventional Taylor series withα=0.

[0072]FIG. 5A˜5B are graphs for comparing approximation errors ofarctan(x) according to the Taylor series expansion method, the Eulerapproximation method, and the approximation method of the presentinvention. For fairness in comparison, all these methods use the sameorder of the highest terms.

[0073] With regard to FIG. 5A, both of the approximation methods, themethod according to the present invention and conventional Taylor seriesexpansion, are expanded up to the 7^(th) term, i.e., N=7. The Eulerapproximation method is obtained using the Taylor series expansion up to3^(rd) term, and by using the Euler transformation formula for the4^(th) to 7^(th) terms. In FIG. 5A, the approximation errorcharacteristics, according to the Euler approximation method, appear tobe better than others for inputs whose absolute value |x| is nearer 1.However, it is shown that the approximation error characteristicsaccording to the present invention is much better than the conventionalTaylor series expansion and the Euler method in overall input areas.

[0074] Referring to FIG. 5B, in the case of N=5 and Δ³, theapproximation error characteristics, according to the Eulerapproximation method, shows that the error is so large that desirableperformance may not be expected. On the other hand, the approximationerror characteristics according to the present invention are not muchdifferent from those in the case of N=7. Therefore, it is understoodthat the approximation method according to the present invention is easyto implement and approximates with superior performance regardless ofthe number of terms N.

[0075] If it is required to have a uniform approximation errorcharacteristic over a wide input range, a method in which the input areais divided into several sub-intervals and the new approximation methodfor series expansion as described above according to the presentinvention is applied to a function in each of the sub-intervals, can beused (referred as a sub-interval divisional approximation method forshort). The approximation error becomes much less with the sub-intervaldivisional approximation method. The sub-interval divisionalapproximation algorithm can prevent exponentially increasingapproximation errors as input x becomes more distant from 0.

[0076]FIG. 6A shows an approximation error characteristic of arctan(x),which is expanded by the sub-interval divisional approximation method,wherein inputs of arctan(x) are divided into 4 sub-intervals, each ofwhich is centered at x=0.75, x=−0.25, x=0.25 and x=0.75, respectively,and arctan(x) in each sub-interval is series-expanded according to thepresent invention. In this sub-interval divisional approximation method,each approximation error characteristic does not necessarily take theform of alternating series as in the approximation algorithm, the inputof which is centered at x=0. However, in each sub-interval, acorresponding arctangent function arctan(x) approximated by Taylorseries expansion still maintains the convergence characteristic. Hence,the approximation error can be reduced in each sub-interval by combiningthe sum of (N−1) terms, expanded by Taylor series expansion, and theN^(th) term multiplied by a weight β, obtained by using the sameprinciple to obtain α as well as considering the sign characteristic.The sub-interval divisional approximation method, to be applied tocorresponding sub-intervals, is represented as follows:

S _(N)(x)=S _(N−1)(x)+βƒ_(N)(x)   (10)

[0077] For each sub-interval, there may be two values of β used. Inother words, since the approximation error function may have thecharacteristic of an alternating series in the right or left directionswith respect to a given center in each of the sub-intervals, eachsub-interval should be further divided into 2 sub-intervals, forexample, [−1, −0.75], [−0.75, −0.5], . . . [0.75, 1], and the optimalvalue of β is calculated in each of 8 sub-intervals. The optimal valueof β in each of the 8 sub-intervals can be calculated in the same manneras that for obtaining α. The sub-interval divisional approximationmethod as described above may be implemented in any processor in thesystem having an operational processing function.

[0078]FIG. 6B shows two comparable approximation error characteristicsof function arctan(x) implemented by Taylor series expansion and by themethod of the present invention, respectively, input intervals in bothcases being sub-divided, and both methods using the same number ofexpansion terms N=4. In FIG. 6B, β's for the 8 sub-intervals areobtained as 0.888, 1.10, 0.445, 1.56, 1.58, 0.465, 1.1, 0.892,respectively. Since the sub-interval divisional approximation approachusing the Euler approximation method cannot be possible, theapproximation error characteristic by the Euler approximation method forcomparison are not shown in FIG. 6B. As shown in FIG. 6B, if an area[−1, 1] (from −1 to 1) is sub-divided into 4 sub-intervals, theapproximation error characteristic according to the present inventionbecomes much less and more uniform over the input area than the errorcharacteristic according to the conventional Taylor method. Thisperformance can be more improved if the input area is divided morenarrowly.

[0079] Accordingly, an approximation method and apparatus for a seriesexpansion of functions, according to the present invention, can reducethe approximation error when expanded with a finite number of terms andhas better applicability than Euler approximation method.

[0080] Now there will be described a carrier recovery process in anOrthogonal Frequency Division Multiplexing (OFDM) system to which anapproximation method of the present invention is suitably applied. Thecarrier recovery process in OFDM system often involves Taylor seriesexpansion of trigonometric functions, to which the present invention maybe applied.

[0081] For carrier frequency offset estimation in OFDM system, thefollowing equation (11) based on the rear part of OFDM symbol and theguard interval is generally used. $\begin{matrix}{\hat{ɛ} = {\frac{1}{2\quad \pi}\arctan \left\{ \frac{\sum\limits_{i = 1}^{L}{{Im}\left( {{y\left( {- i} \right)}y*\left( {L - i} \right)} \right)}}{\sum\limits_{i = 1}^{L}{{Re}\left( {{y\left( {- i} \right)}y*\left( {L - i} \right)} \right)}} \right\}}} & (11)\end{matrix}$

[0082] where, Re and Im represent the real part and imaginary part,respectively, of a complex number, y(i) represents an i-th receivedsample, L is the fast fourier transformation (FFT) size, and {circumflexover (ε)} is the estimated and normalized carrier frequency offset ofΔ{circumflex over (ƒ)}T . When an arctan(x) in equation (1) isapproximated by Taylor series expansion, the biggest approximation erroroccurs at x=±1, wherein {circumflex over (ε)} is {circumflex over(ε)}=⅛, ±⅜. If it is assumed that the symbol period T=3.2 μsec as inIEEE 802.11a, the biggest estimation error by Taylor series expansionoccurs at Δƒ=×39.1 KHz, and ±117.2 KHz. In addition, if the seriesexpansion of arctan(x) resulted in an approximation error of A° (° meansdegree), this means that the normalized frequency offset of A/360 occursregardless of the estimated frequency offset. Since an OFDM system is sosusceptible to the frequency offset that even a small frequency offsetcan cause a substantial bit-error-rate (BER) degradation due tointer-carrier interference (ICI), the frequency offset must be kept to aminimum.

[0083] Such an estimated frequency offset$\hat{ɛ} = {{\Delta \quad \hat{f}T} = \frac{\Delta \quad \hat{\varpi}T}{2\quad \pi}}$

[0084] is compensated by using a trigonometric expansion formula likethe following equation (12) to estimate the phase rotation angle of thek-th sample with sample interval T_(s), that is kΔwT_(s) in radians,like the following equation (13). Equation (13) is derived from theapproximation principle of the present invention. In following equations(12) and (13), it is noted that initial estimation frequency offsetvalue (for k=1) is preferably estimated to prevent cumulative offsets.

sin(α±β)=sin(α)cos(β)±cos(α)sin(β)

cos(α±β)=cos(α)cos(β)±sin(α)sin(β)   (12)

[0085] $\begin{matrix}\begin{matrix}{{{{For}\quad k} = 1},{{\cos \left( {\Delta \quad \hat{\omega}T_{s}} \right)} = {\sum\limits_{n = 0}^{N}{\left( {- 1} \right)^{n}\frac{\Delta \quad \hat{\omega}T_{s}^{2n}}{\left( {2n} \right)!}}}}} \\{\quad {{\sin \left( {\Delta \quad \hat{\omega}T_{s}} \right)} = {\sum\limits_{n = 0}^{N}{\left( {- 1} \right)^{n}\frac{\Delta \quad \hat{\omega}T_{s}^{({{2n} + 1})}}{\left( {{2n} + 1} \right)!}}}}} \\{{{{For}\quad k} \geq 2},{{\cos \left( {k\quad \Delta \quad \hat{\omega}T_{s}} \right)} = {\cos \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} + {\Delta \quad \hat{\omega}T_{s}}} \right)}}} \\{\quad {= {{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}T_{s}} \right)}} -}}} \\{\quad {{\sin \left( {\left( {k - 1} \right)\quad \Delta \quad \hat{\omega}T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}} \\{\quad {{\sin \left( {k\quad \Delta \quad \hat{\omega}T_{s}} \right)} = {\sin \left( {{\left( {k - 1} \right)\Delta \hat{\omega}T_{s}} + {\Delta \quad \hat{\omega}T_{s}}} \right)}}} \\{\quad {= {{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}T_{s}} \right)}} +}}} \\{\quad {{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}T_{s}} \right)}}}\end{matrix} & (13)\end{matrix}$

[0086] In equation (13), in order to compensate for the phase rotationfor the first sample of k=1, the series expansion approximationalgorithm is used to obtain cos(ΔŵT_(s)) and sin(ΔŵT_(s)). For samplesfollowing the first one, k>2, the results of the trigonometric functionsobtained for previous sample and for the first sample of k=1 are used toobtain corresponding values of trigonometric functions expanded as inequation (12). In most OFDM system, pilots are inserted in a data areadespite the data efficiency loss to prevent phase rotation due to thefrequency offset estimation error. However, in an OFDM system accordingto the present invention, since a series expansion approximationfunction of the present invention as described above is used to minimizethe frequency offset estimation error, it is allowed to eliminate such adata efficiency loss factor.

[0087]FIG. 7A˜7C are constellation diagrams of the Quadrature AmplitudeModulation-Orthogonal Frequency Division Multiplexing (QAM-OFDM),observed for 10 OFDM symbols, according to a conventional seriesexpansion approximation method and the approximation method of thepresent invention, respectively. Referring to FIG. 5 again, when theapproximation method of the present invention was used at x=−0.94, theworst approximation error occurred. FIG. 7A and FIG. 7B have beencreated by performing Taylor series expansion of a function with afinite number of terms, N=7 and N=8, and resulting approximation errors0.65° and 0.52°, respectively. FIG. 7C results from performing theapproximation method according to the present invention. In FIG. 7C,when x=−0.94, the approximation error |E_(N)(x)| is 0.047°. From theFIG. 7C, it is shown that QAM-OFDM constellation is placed in optimalposition without phase rotation.

[0088] As described above, a new approximation method for Taylor seriesexpansion according to the present invention enables a simpleapproximation to be made, is easy to be implemented, and minimizes theapproximation error.

[0089] While this invention has been particularly shown and describedwith reference to preferred embodiments thereof, it will be understoodby those skilled in the art that various changes in form and detailmaybe made therein without departing from the spirit and scope of theinvention as defined by the appended claims.

What is claimed is:
 1. An approximation method for a series expansion ofan input function with a finite number of terms N to minimize anapproximation error, comprising: expanding the input function in Taylorseries up to an (N−1)-th term to obtain a first expansion result;expanding the input function in Taylor series up to an N-th term toobtain a second expansion result; multiplying the first expansion resultby a predetermined weight α to obtain a multiplication result; combiningthe multiplication result and the second expansion result to obtain acombined result; and dividing the combined result by(α+1).
 2. The methodof claim 1, wherein a is greater than 0 and no greater than
 1. 3. Anapproximation method for a series expansion of an input function with afinite number of terms N to minimize an approximation error, comprising:expanding the input function in Taylor series up to an (N−1)-th term toobtain an expansion result; multiplying an N-th term of the expansionresult by a predetermined weight value to obtain a multiplicationresult; and combining the expansion result and the multiplication resultto obtain an approximation function ƒ for the series expansion of theinput function.
 4. The method of claim 3, wherein the predeterminedweight value is$\frac{\left( {- 1} \right)^{N}}{\left( {\alpha + 1} \right)},$

for 0<α≦1.
 5. The method of claim 4, wherein a value α obtained for acorresponding respective N is selected so as to minimize a maximum ofthe approximation error.
 6. The method of claim 4, wherein the value ofα is obtained by: (a) selecting a minimum input in a given input x area;(b) calculating the approximation function ƒ for the input with thefinite number of terms N; (c) obtaining and storing an error |E_(N,x)|by subtracting the approximation function ƒ from a nominal functionvalue of the input x; (d) determining whether the input x has reached amaximum value in the given input x area; (e) adding a predeterminedincrement ξ to the input x if the input x has not yet reached themaximum value, and repeating steps (b), (c) and (d); (f) selecting amaximum error value among all the stored errors of |E_(N,x)| for allinputs when x has reached a maximum value; and (g) searching the value αto minimize the maximum error value, and storing the value α as theweight value for a corresponding N.
 7. The method of claim 3, whereinthe value of α is obtained by: (a) selecting a minimum input in a giveninput x area; (b) calculating the approximation function ƒ for the inputwith the finite number of terms N; (c) obtaining and storing an error|E_(N,x)| by subtracting the approximation function ƒ from a nominalfunction value of the input x; (d) determining whether the input x hasreached a maximum value in the given input x area; (e) adding apredetermined increment ξ to the input x if the input x has not yetreached the maximum value, and repeating steps (b), (c) and (d); (f)selecting a maximum error value among all the stored errors of |E_(N,x)|for all inputs when x has reached a maximum value; and (g) searching thevalue α to minimize the maximum error value, and storing the value α asthe weight value for a corresponding N.
 8. An approximation method for aseries expansion of an input function with a finite number of terms N tominimize an approximation error, comprising: dividing a whole input areainto several predetermined sub-intervals: expanding the input functionin Taylor series up to an (N−1)-th term in each of the sub-intervals toobtain a series expansion for each sub-interval; multiplying an N-thterm of the series expansion of the input function with a predeterminedfirst weight for inputs on a left side of a center for said each of thesub-intervals; multiplying an N-th term of the series expansion with apredetermined second weight for inputs on a right side of the center forsaid each of the sub-intervals; and combining the series expansion andthe multiplied N-th term with the predetermined first and second weightsto obtain an approximation of the input function in said each of thesub-intervals.
 9. The method of claim 8, wherein the predetermined firstand second weights on the left and right side, respectively, in saideach of the sub-intervals are selected to minimize a maximum errorbetween the approximation of the input function with the finite numberof terms N and a nominal value of the input function over all inputs incorresponding sub-intervals.
 10. A method for compensating a carrierfrequency offset in an orthogonal frequency division multiplexing (OFDM)system, comprising: estimating the carrier frequency offset {circumflexover (ε)} by using a series expansion of an arctangent functionarctan(x); using the estimated carrier frequency offset to obtain aphase rotation value for a first input sample of k=1, wherein sin(2πê)and cos(2πê) are series-expanded to minimize an approximation error;using a phase rotation value for a previous input sample including k=1to obtain a phase rotation value for a subsequent input sample; andcompensating the phase rotation values for all input samples.
 11. Themethod of claim 10, wherein the estimated carrier frequency offset{circumflex over (ε)} is represented by${\hat{ɛ} = {\frac{1}{2\quad \pi}\arctan \left\{ \frac{\sum\limits_{i = 1}^{L}{{Im}\left( {{y\left( {- i} \right)}y*\left( {L - i} \right)} \right)}}{\sum\limits_{i = 1}^{L}{{Re}\left( {{y\left( {- i} \right)}y*\left( {L - i} \right)} \right)}} \right\}}},$

where Re and Im represent a real part and an imaginary part,respectively, of a complex number, y(i) represents an i-th receivedsample, L is a fast fourier transformation (FFT) size, and {circumflexover (ε)} is an estimated and normalized carrier frequency offset ofΔ{circumflex over (ƒ)}T.
 12. The method of claim 11, wherein the phaserotation value for a k-th sample is calculated by: $\begin{matrix}{{{{For}\quad k} = 1},{{\cos \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sum\limits_{n = 0}^{N}}^{{({- 1})}^{n}\frac{\Delta \quad \hat{\omega}\quad T_{s}^{2n}}{{({2n})}!}}}} \\{\quad {{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sum\limits_{n = 0}^{N}}^{{({- 1})}^{n}\frac{\Delta \quad \hat{\omega}\quad T_{s}^{({{2n} + 1})}}{{({{2n} + 1})}!}}}} \\\begin{matrix}{{{{For}\quad k} \geq 2},{{\cos \left( {k\quad \Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\cos \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} + {\Delta \quad \hat{\omega}\quad T_{s}}} \right)}}} \\{= {{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}\cos \quad \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} -}} \\{{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}}\end{matrix} \\\begin{matrix}{\quad {{\sin \left( {k\quad \Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sin \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} + {\Delta \quad \hat{\omega}\quad T_{s}}} \right)}}} \\{= {{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}} +}} \\{{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}}\end{matrix}\end{matrix}$


13. The method of claim 10, wherein the phase rotation value for a k-thsample is calculated by: $\begin{matrix}{{{{For}\quad k} = 1},{{\cos \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sum\limits_{n = 0}^{N}}^{{({- 1})}^{n}\frac{\Delta \quad \hat{\omega}\quad T_{s}^{2n}}{{({2n})}!}}}} \\{\quad {{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sum\limits_{n = 0}^{N}}^{{({- 1})}^{n}\frac{\Delta \quad \hat{\omega}\quad T_{s}^{({{2n} + 1})}}{{({{2n} + 1})}!}}}} \\\begin{matrix}{{{{For}\quad k} \geq 2},{{\cos \left( {k\quad \Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\cos \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} + {\Delta \quad \hat{\omega}\quad T_{s}}} \right)}}} \\{= {{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}\cos \quad \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} -}} \\{{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}}\end{matrix} \\\begin{matrix}{\quad {{\sin \left( {k\quad \Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sin \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} + {\Delta \quad \hat{\omega}\quad T_{s}}} \right)}}} \\{= {{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}} +}} \\{{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}}\end{matrix}\end{matrix}$


14. An approximation system for a series expansion of an input functionwith a finite number of terms N to minimize an approximation error,comprising: an operational processing unit which expands the inputfunction in Taylor series up to an (N−1)-th term to obtain a firstexpansion result, expands the input function in Taylor series up to anN-th term to obtain a second expansion result, multiplies the firstexpansion result by a predetermined weight α to obtain a multiplicationresult, combines the multiplication result and the second expansionresult to obtain a combined result, and divides the combined result by(α+1).
 15. The system of claim 14, wherein α is greater than 0 and nogreater than
 1. 16. The system of claim 14, wherein α obtained for acorresponding respective N is selected so as to minimize a maximumapproximation error.
 17. An approximation system for a series expansionof an input function with a finite number of terms N to minimize anapproximation error, comprising: an operational processing unit whichexpands the input function in Taylor series up to an (N−1)-th term toobtain an expansion result, multiplies an N-th term of the expansionresult by a predetermined weight value to obtain a multiplicationresult, and combines the expansion result and the multiplication resultto obtain an approximation function ƒ for the series expansion function.18. The system of claim 17, wherein the predetermined weight value is$\frac{\left( {- 1} \right)^{N}}{\left( {\alpha + 1} \right)}$

for 0<α≦1.
 19. The system of claim 18, wherein α obtained forcorresponding respective N is selected to minimize a maximumapproximation error.
 20. The system of claim 19, wherein α is obtainedby: (a) selecting a minimum input in a given input x area; (b)calculating the approximation function ƒ for the input function with thefinite number of terms N (c) obtaining and storing an error E_(N,x) bysubtracting approximation function ƒ from a nominal function value ofthe input x; (d) determining whether the input x has reached a maximumvalue in the given input x area, adding a predetermined increment ξ to xwhen x has not yet reached the maximum value, and repeating steps (b),(c) and (d); (e) selecting a maximum error value among all the storederrors E_(N,x) for all inputs when x has reached a maximum value; and(f) searching α to minimize the maximum error value, and storing α asthe weight value for a corresponding N.
 21. The system of claim 17,wherein a obtained for corresponding respective N is selected tominimize a maximum approximation error.
 22. An orthogonal frequencydivision multiplexing (OFDM) system for compensating a carrier frequencyoffset, comprising: an estimator for estimating the carrier frequencyoffset ê by using a series expansion of a function arctan(x); a firstphase rotation calculator for using the estimated carrier frequencyoffset to obtain a phase rotation value for a first input sample of k=1,wherein sin(2πê) and cos(2πê) are series-expanded to minimize anapproximation error; a second phase rotation calculator for using aphase rotation value for a previous input sample including k=1 to obtaina phase rotation value for a subsequent input sample; and a compensatorfor compensating the phase rotation values for all input samples. 23.The system of claim 22, wherein the estimated carrier frequency offset êis represented by${\hat{ɛ} = {\frac{1}{2\pi}\arctan \left\{ \frac{\sum\limits_{i = 1}^{L}\quad {{Im}\left( {{y\left( {- i} \right)}{y^{*}\left( {L - i} \right)}} \right)}}{\sum\limits_{i = 1}^{L}\quad {{Re}\left( {{y\left( {- i} \right)}{y^{*}\left( {L - i} \right)}} \right)}} \right\}}},$

where Re and Im represent a real part and an imaginary part,respectively, of a complex number, y(i) represents an i-th receivedsample, L is a fast fourier transformation (FFT) size, and ê is anestimated and normalized carrier frequency offset of Δ{circumflex over(ƒ)}T.
 24. The method of claim 23, wherein the phase rotation value fora k-th sample is calculated by $\begin{matrix}{{{{For}\quad k} = 1},{{\cos \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sum\limits_{n = 0}^{N}}^{{({- 1})}^{n}\frac{\Delta \quad \hat{\omega}\quad T_{s}^{2n}}{{({2n})}!}}}} \\{\quad {{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sum\limits_{n = 0}^{N}}^{{({- 1})}^{n}\frac{\Delta \quad \hat{\omega}\quad T_{s}^{({{2n} + 1})}}{{({{2n} + 1})}!}}}} \\\begin{matrix}{{{{For}\quad k} \geq 2},{{\cos \left( {k\quad \Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\cos \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} + {\Delta \quad \hat{\omega}\quad T_{s}}} \right)}}} \\{= {{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}\cos \quad \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} -}} \\{{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}}\end{matrix} \\\begin{matrix}{\quad {{\sin \left( {k\quad \Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sin \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} + {\Delta \quad \hat{\omega}\quad T_{s}}} \right)}}} \\{= {{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}} +}} \\{{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}}\end{matrix}\end{matrix}$


25. The method of claim 22, wherein the phase rotation value for a k-thsample is calculated by $\begin{matrix}{{{{For}\quad k} = 1},{{\cos \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sum\limits_{n = 0}^{N}}^{{({- 1})}^{n}\frac{\Delta \quad \hat{\omega}\quad T_{s}^{2n}}{{({2n})}!}}}} \\{\quad {{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sum\limits_{n = 0}^{N}}^{{({- 1})}^{n}\frac{\Delta \quad \hat{\omega}\quad T_{s}^{({{2n} + 1})}}{{({{2n} + 1})}!}}}} \\\begin{matrix}{{{{For}\quad k} \geq 2},{{\cos \left( {k\quad \Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\cos \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} + {\Delta \quad \hat{\omega}\quad T_{s}}} \right)}}} \\{= {{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}\cos \quad \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)} -}} \\{{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}}\end{matrix} \\\begin{matrix}{\quad {{\sin \left( {k\quad \Delta \quad \hat{\omega}\quad T_{s}} \right)} = {\sin \left( {{\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} + {\Delta \quad \hat{\omega}\quad T_{s}}} \right)}}} \\{= {{{\sin \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\cos \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}} +}} \\{{{\cos \left( {\left( {k - 1} \right)\Delta \quad \hat{\omega}\quad T_{s}} \right)}{\sin \left( {\Delta \quad \hat{\omega}\quad T_{s}} \right)}}}\end{matrix}\end{matrix}$